Editing Lambert W function (section)

== Generalizations ==
The standard Lambert {{mvar|W}} function expresses exact solutions to ''transcendental algebraic'' equations (in {{mvar|x}}) of the form:
{{NumBlk||<math display="block"> e^{-c x} = a_0 (x-r)</math>|{{EquationRef|1}}}}
where {{math|''a''<sub>0</sub>}}, {{mvar|c}} and {{mvar|r}} are real constants. The solution is
<math display="block"> x = r + \frac{1}{c} W\left( \frac{c\,e^{-c r}}{a_0} \right). </math>
Generalizations of the Lambert {{mvar|W}} function<ref>{{cite journal |first1=T. C. |last1=Scott |first2=R. B. |last2=Mann |year=2006 |title=General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert ''W'' Function |journal=AAECC (Applicable Algebra in Engineering, Communication and Computing) |volume=17 |issue=1 |pages=41–47 |doi=10.1007/s00200-006-0196-1 |arxiv=math-ph/0607011 |bibcode=2006math.ph...7011S |last3=Martinez Ii |first3=Roberto E. |s2cid=14664985 }}</ref><ref>{{cite journal |first1=T. C. |last1=Scott |first2=G. |last2=Fee |first3=J. |last3=Grotendorst |year=2013 |title=Asymptotic series of Generalized Lambert ''W'' Function |journal=SIGSAM (ACM Special Interest Group in Symbolic and Algebraic Manipulation) |volume=47 |issue=185 |pages=75–83|doi=10.1145/2576802.2576804|s2cid=15370297 |url=http://www.sigsam.org/cca/issues/issue185.html}}</ref><ref>{{cite journal |first1=T. C. |last1=Scott |first2=G. |last2=Fee |first3=J. |last3=Grotendorst|first4=W.Z. |last4=Zhang| year=2014|title=Numerics of the Generalized Lambert ''W'' Function|journal=SIGSAM |volume=48 |issue=1/2| pages=42–56| doi=10.1145/2644288.2644298| s2cid=15776321 |url=http://www.sigsam.org/cca/issues/issue188.html}}</ref> include:

<ul>
<li>An application to [[general relativity]] and [[quantum mechanics]] ([[Quantum gravity#The dilaton|quantum gravity]]) in lower dimensions, in fact a link (unknown prior to 2007<ref>{{cite journal |first1=P. S. |last1=Farrugia |first2=R. B. |last2=Mann |first3=T. C. |last3=Scott |year=2007 |title=''N''-body Gravity and the Schrödinger Equation |journal=Class. Quantum Grav. |volume=24 |issue=18 |pages=4647–4659 |doi=10.1088/0264-9381/24/18/006 |arxiv=gr-qc/0611144 |bibcode=2007CQGra..24.4647F |s2cid=119365501 }}</ref>) between these two areas, where the right-hand side of ({{EquationNote|1}}) is replaced by a quadratic polynomial in {{math|''x''}}:
{{NumBlk||<math display="block">e^{-c x} = a_0 \left(x-r_1 \right) \left(x-r_2 \right),</math>|{{EquationRef|2}}}}
where {{math|''r''<sub>1</sub>}} and {{math|''r''<sub>2</sub>}} are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function which has a single argument {{mvar|x}} but the terms like {{math|''r<sub>i</sub>''}} and {{math|''a''<sub>0</sub>}} are parameters of that function. In this respect, the generalization resembles the [[hypergeometric]] function and the [[Meijer G-function|Meijer {{mvar|G}} function]] but it belongs to a different ''class'' of functions. When {{math|1=''r''<sub>1</sub> = ''r''<sub>2</sub>}}, both sides of ({{EquationNote|2}}) can be factored and reduced to ({{EquationNote|1}}) and thus the solution reduces to that of the standard {{mvar|W}} function. Equation ({{EquationNote|2}}) expresses the equation governing the [[dilaton]] field, from which is derived the metric of the [[R = T model|{{math|1=''R'' = ''T''}}]] or ''lineal'' two-body gravity problem in 1&nbsp;+&nbsp;1 dimensions (one spatial dimension and one time dimension) for the case of unequal rest masses, as well as the eigenenergies of the quantum-mechanical [[Delta potential#Double Delta Potential|double-well Dirac delta function model]] for ''unequal'' charges in one dimension.
</li>
<li>Analytical solutions of the eigenenergies of a special case of the quantum mechanical [[Euler's three-body problem|three-body problem]], namely the (three-dimensional) [[hydrogen molecule-ion]].<ref>{{cite journal |first1=T. C. |last1=Scott |first2=M. |last2=Aubert-Frécon |first3=J. |last3=Grotendorst |year=2006 |title=New Approach for the Electronic Energies of the Hydrogen Molecular Ion |journal=Chem. Phys. |volume=324 |issue=2–3 |pages=323–338 |doi=10.1016/j.chemphys.2005.10.031 |arxiv=physics/0607081 |bibcode=2006CP....324..323S |citeseerx=10.1.1.261.9067 |s2cid=623114 }}</ref> Here the right-hand side of ({{EquationNote|1}}) is replaced by a ratio of infinite order polynomials in {{mvar|x}}:
{{NumBlk||<math display="block"> e^{-c x} = a_0 \frac{\displaystyle \prod_{i=1}^\infty (x-r_i )}{\displaystyle \prod_{i=1}^\infty (x-s_i)}</math>|{{EquationRef|3}}}}
where {{math|''r''<sub>''i''</sub>}} and {{math|''s''<sub>''i''</sub>}} are distinct real constants and {{mvar|x}} is a function of the eigenenergy and the internuclear distance {{mvar|R}}. Equation ({{EquationNote|3}}) with its specialized cases expressed in ({{EquationNote|1}}) and ({{EquationNote|2}}) is related to a large class of [[delay differential equation]]s. [[G. H. Hardy]]'s notion of a "false derivative" provides exact multiple roots to special cases of ({{EquationNote|3}}).<ref>{{cite journal |first1=Aude |last1=Maignan |first2=T. C. |last2=Scott |year=2016|title=Fleshing out the Generalized Lambert ''W'' Function| journal=SIGSAM |volume=50 |issue=2|pages=45–60|doi=10.1145/2992274.2992275|s2cid=53222884 }}</ref>
</li>
</ul>
Applications of the Lambert {{mvar|W}} function in fundamental physical problems are not exhausted even for the standard case expressed in ({{EquationNote|1}}) as seen recently in the area of [[atomic, molecular, and optical physics]].<ref>{{cite journal |first1=T. C. |last1=Scott |first2=A. |last2=Lüchow |first3=D. |last3=Bressanini |first4=J. D. III |last4=Morgan |year=2007 |title=The Nodal Surfaces of Helium Atom Eigenfunctions |journal=[[Physical Review|Phys. Rev. A]] |volume=75 |issue=6 |pages=060101 |doi=10.1103/PhysRevA.75.060101 |bibcode=2007PhRvA..75f0101S |hdl=11383/1679348 |url=https://irinsubria.uninsubria.it/bitstream/11383/1679348/1/2007-scott-pra2007.pdf |archive-url=https://web.archive.org/web/20170922054740/https://irinsubria.uninsubria.it/bitstream/11383/1679348/1/2007-scott-pra2007.pdf |archive-date=2017-09-22 |url-status=live |hdl-access=free }}</ref>